In this article, we derive polynomial-dimension theoretical assurances
of unadjusted LMC algorithm for a family of potentials that are \(\alpha \) -mixture
weakly smooth and isoperimetric (i.e. log Sobolev, Poincaré, and Talagrand).
In addition, we also investigate the family of potential which is
non-strongly convex outside the ball and 2-dissipative. The analysis
we proposed is an extension of the recently published paper [1]}
in combination with the convexification of non-convex domain [2]}.
There are a number of valuable potential directions which one can
explore, among them we speculate some here. It is potential to broaden
our results to apply underdamped LMC or higher order LMC to these
class of potential while the computational complexity remains polynomial
dependence on \(d\) . Another fascinating question is whether it is
feasible to sampling from distributions with non-smooth and totally
non-convex structure and integrate into derivative-free LMC algorithm.
